Geometry & Topology

End reductions, fundamental groups, and covering spaces of irreducible open 3–manifolds

Robert Myers

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Suppose M is a connected, open, orientable, irreducible 3–manifold which is not homeomorphic to . Given a compact 3–manifold J in M which satisfies certain conditions, Brin and Thickstun have associated to it an open neighborhood V called an end reduction of M at J. It has some useful properties which allow one to extend to M various results known to hold for the more restrictive class of eventually end irreducible open 3–manifolds.

In this paper we explore the relationship of V and M with regard to their fundamental groups and their covering spaces. In particular we give conditions under which the inclusion induced homomorphism on fundamental groups is an isomorphism. We also show that if M has universal covering space homeomorphic to , then so does V.

This work was motivated by a conjecture of Freedman (later disproved by Freedman and Gabai) on knots in M which are covered by a standard set of lines in .

Article information

Geom. Topol., Volume 9, Number 2 (2005), 971-990.

Received: 14 July 2004
Revised: 18 May 2005
Accepted: 18 May 2005
First available in Project Euclid: 20 December 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M10: Covering spaces
Secondary: 57N10: Topology of general 3-manifolds [See also 57Mxx] 57M27: Invariants of knots and 3-manifolds

3–manifold end reduction covering space


Myers, Robert. End reductions, fundamental groups, and covering spaces of irreducible open 3–manifolds. Geom. Topol. 9 (2005), no. 2, 971--990. doi:10.2140/gt.2005.9.971.

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  • I Agol, Tameness of hyperbolic 3–manifolds.
  • M G Brin, T L Thickstun, Open, \irr \tm s which are end 1–movable, Topology 26 (1987) 211–233
  • M G Brin, T L Thickstun, 3–manifolds which are end 1–movable, Memoirs Amer. Math. Soc. 81 (1989)
  • E M Brown, C D Feustel, On properly embedding planes in arbitrary 3–manifolds, Proc. Amer. Math. Soc. 94 (1985) 173–178
  • D C Calegari, D Gabai, Shrinkwrapping and the taming of hyperbolic 3–manifolds.
  • A V Chernavskii, Local contractibility of the group of homeomorphisms of a manifold, Mat. Sb. (N.S.) 79 (121) (1969) 307–356
  • R D Edwards, R C Kirby, Deformations of spaces of imbeddings, Ann. Math. (2) 93 (1971) 63–88
  • M Freedman, D Gabai, Covering a nontaming knot by the unlink, preprint available at or at
  • M Freedman, V Krushkal, Notes on ends of hyperbolic 3–manifolds, from: “Proc. 13th Annual Workshop in Geometric Topology, Colorado College, Colorado Springs, CO (June 13–15, 1996)” 1–15
  • H Freudenthal, Neuaufbau der Endentheorie, Ann. of Math. (2) 43 (1942) 261–279
  • J Hempel, 3–manifolds, Ann. of Math. Studies 86, Princeton (1976)
  • W Jaco, Lectures on three-manifold topology, CBMS Regional Conference Series in Math. 43, Amer. Math. Soc. (1980)
  • D R McMillan Jr, Cartesian products of contractible open manifolds, Bull. Amer. Math. Soc. 67 (1961) 510–514
  • D R McMillan Jr, Compact, acyclic subsets of three-manifolds, Michigan Math. J. 16 (1969) 129–136
  • D R McMillan Jr, Acyclicity in three-manifolds, Bull. Amer. Math. Soc. 76 (1970) 942–964
  • R Myers, End reductions and covering translations of contractible open \tm s.
  • P Scott, Compact submanifolds of 3–manifolds, J. London Math. Soc. (2) 7 (1973) 246–250